It occurred to me that one ought to be able to fit a circle to any parabola, thus creating a smooth two-dimensional shape more or less like a plane section along the long axis of an egg. In fact, the resulting graph is not merely smooth, but continuous, which makes it interesting, since the continuity derives from the seamless fusion of two functions.
I decided to work with the basic parabolic equation, y = x2, since all of the others are transformations, and similar principles would apply. Besides, the curve I was most interested in was parabolic at its small end and circular at its large, and I wanted the bottom of that curve to pass through (0, 0). (It’s possible to construct a closed curve circular at each end and parabolic on the sides by calculating a smaller and a larger circle for each parabola and replacing the bottom of the parabola with the bottom of the smaller circle.)
In order to make the fusion seamless, it would be necessary to join the two curves precisely where they had the same slope, or tangent—in other words, where the derivatives of the two functions were equal.
Since the absolute value of Dx for the function y = x2 is always rising but never reaches infinity, and since the absolute value of Dx for the general equation of a circle can only reach infinity when x = the radius of the implied circle, the fusion of the two curves must always occur before that point. In other words, no matter how great the absolute value of x, you can always close the loop of the parabola by fusing it to a circle.
It develops that a circle fused to a parabola at x and –x will have its center at (0, x2 + ½ ) and a radius of (x2 + ¼ )1/2 (absolute value). From this it’s obvious that the radius of the circle can never be equal to or greater than the height of the center above (0,0), which verifies the assertion that one can close a parabola of any size with a circle. The single exception occurs when x = 0. When x = 0, the parabola disappears completely, leaving only a circle of radius ½ centered at (0, ½ )—the bottom of the curve still passing through (0,0).
The greater the absolute value of x, the larger the circle surmounting the parabola, obviously enough, and the more elongated the egg shape. I had been thinking all along of rotating the curve on the y-axis to generate a more or less ovoid three-dimensional surface. Now I wondered what the most pleasing proportions for such an object would be—in other words, what two-dimensional closed curve would produce the most pleasant proportions when rotated.
I appealed to the Golden Mean, and decided to find what value of x would produce a curve whose major axis (along the y-axis) was approximately 1.61803399 times the length of the diameter of its surmounting circle (twice the radius)—in short, a curve approximately 1/.61803399 times as tall as it is wide at the “shoulders.”
It turns out that to produce the desired curve, x must = approximately 2.05817103, the radius of the circle = approximately 1/.61803399 + ½ (or 2.11803399), and the major axis = approximately 6.85410197.
One could argue whether those are indeed the most pleasing proportions, but regardless, I got a good title out of it for this discourse.